Theorem foco2 7113

Description: If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.)

Assertion

Ref	Expression
foco2	⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) → 𝐹:𝐵–onto→𝐶)

Proof of Theorem foco2

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		foelrn 7111	. . . . . 6 ⊢ (((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃𝑧 ∈ 𝐴 𝑦 = ((𝐹 ∘ 𝐺)‘𝑧))
2		ffvelcdm 7085	. . . . . . . . 9 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → (𝐺‘𝑧) ∈ 𝐵)
3		fvco3 6991	. . . . . . . . 9 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧)))
4		fveq2 6891	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺‘𝑧) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑧)))
5	4	rspceeqv 3629	. . . . . . . . 9 ⊢ (((𝐺‘𝑧) ∈ 𝐵 ∧ ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧))) → ∃𝑥 ∈ 𝐵 ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥))
6	2, 3, 5	syl2anc 583	. . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥))
7		eqeq1 2731	. . . . . . . . 9 ⊢ (𝑦 = ((𝐹 ∘ 𝐺)‘𝑧) → (𝑦 = (𝐹‘𝑥) ↔ ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥)))
8	7	rexbidv 3173	. . . . . . . 8 ⊢ (𝑦 = ((𝐹 ∘ 𝐺)‘𝑧) → (∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐵 ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘𝑥)))
9	6, 8	syl5ibrcom 246	. . . . . . 7 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑦 = ((𝐹 ∘ 𝐺)‘𝑧) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
10	9	rexlimdva 3150	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → (∃𝑧 ∈ 𝐴 𝑦 = ((𝐹 ∘ 𝐺)‘𝑧) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
11	1, 10	syl5 34	. . . . 5 ⊢ (𝐺:𝐴⟶𝐵 → (((𝐹 ∘ 𝐺):𝐴–onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
12	11	impl 455	. . . 4 ⊢ (((𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥))
13	12	ralrimiva 3141	. . 3 ⊢ ((𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) → ∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥))
14	13	anim2i 616	. 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ (𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶)) → (𝐹:𝐵⟶𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
15		3anass 1093	. 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) ↔ (𝐹:𝐵⟶𝐶 ∧ (𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶)))
16		dffo3 7106	. 2 ⊢ (𝐹:𝐵–onto→𝐶 ↔ (𝐹:𝐵⟶𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐵 𝑦 = (𝐹‘𝑥)))
17	14, 15, 16	3imtr4i 292	1 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) → 𝐹:𝐵–onto→𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∀wral 3056  ∃wrex 3065   ∘ ccom 5676  ⟶wf 6538  –onto→wfo 6540  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550

This theorem is referenced by:  fcoresfo  46376